Hilbert-Schmidt Expansion Research Articles

Deeply quasi-bound state in single- and double- $\bar{K}$ K ¯ nuclear clusters

New calculations of the quasi-bound state positions in $K^{-}K^{-}pp$ kaonic nuclear cluster are performed using non-relativistic four-body Faddeev-type equations in AGS form. The corresponding separable approximation for the integral kernels in the three- and four-body kaonic clusters is obtained by using the Hilbert-Schmidt expansion procedure. Different phenomenological models of $\bar{K}N-\pi\Sigma$ potentials with one- and two-pole structure of $\Lambda$(1405) resonance and separable potential models for $\bar{K}$-$\bar{K}$ and nucleon-nucleon interactions, are used. The dependence of the resulting four-body binding energy on models of $\bar{K}N-\pi\Sigma$ interaction is investigated. We obtained the binding energy of the $K^{-}K^{-}pp$ quasi-bound state $\sim$ 80-94 MeV with the phenomenological $\bar{K}N$ potentials. The width is about $\sim$ 5-8 MeV for the two-pole models of the interaction, while the one-pole potentials give $\sim$ 24-31 MeV width.

Coupled-channels Faddeev AGS calculation of K-ppn and K-ppp quasi-bound states

Using separable $\bar{K}N-\pi\Sigma$ potentials in the Faddeev equations, we calculated the binding energies and widths of the $K^{-}pp$, $K^{-}ppn$ and $K^{-}ppp$ quasi-bound states on the basis of three- and four-body Alt-Grassberger-Sandhas equations in the momentum representation. One- and two-pole version of $\bar{K}N-\pi\Sigma$ interaction are considered and the dependence of the resulting few-body energy on the two-body $\bar{K}N-\pi\Sigma$ potential was investigated. The $s$-wave [3+1] and [2+2] sub-amplitudes are obtained by using the Hilbert-Schmidt expansion procedure for the integral kernels. As a result, we found a four-body resonance of the $K^{-}ppn$ and $K^{-}ppp$ quasi-bound states with a binding energy in the range $B_{K^{-}ppn}\sim{55-70}$ and $B_{K^{-}ppp}\sim{90-100}$ MeV, respectively. The calculations yielded full width of $\Gamma_{K^{-}ppn}\sim{16-20}$ and $\Gamma_{K^{-}ppp}\sim{7-20}$ MeV.

Integral equations in the Fock-Bargmann space and the generator-coordinate method

In the generator-coordinate method, the norm kernel built as a Slater determinant of the Brink-Bloch orbitals plays a role of unity for the complete set of orthonormalized states defined in the Fock-Bargmann space and satisfying the Pauli exclusion principle. The closure condition is then written as the Hilbert-Schmidt expansion of the norm kernel and can be derived by a simple algebraic procedure. A basis of eigenstates of the norm kernel is shown to be a powerful tool in resonating-group calculations.

Faddeev-Born-Oppenheimer equations for molecular three-body systems: Application to H2 +

A non variational molecular like approach is developed for the three-body problem based on the Faddeev equations. Considering a system of two identical heavy particles (atomic nuclei) and a light one (electron) we study the adiabatic limit of the corresponding Faddeev equation in the absence of interaction between the heavy particles and using general heavy-light potentials that are represented in a separable form through Sturmian functions. The resulting rotationally invariant Faddeev two-center eigenfunctions are used to formulate an ansatz for the solution of the full Hamiltonian where all three particles interact. A set of coupled differential Born-Oppenheimer like equations is obtained for the movement of the heavy particles. Numerical calculations are shown for the 1sσg, 2sσg, 3dσg and 2pσu electronic states in 2 + . The resulting molecular energy curves converge to the exact ones when up to thirty six terms are used in the Hilbert-Schmidt expansion of the Coulomb potential. The non-crossing rule for 2sσg and 3dσg curves is verified.

Infinite-rank separable expansion for the three-bodyTmatrix at energies in the continuous spectrum region

A method for expanding a three-particle T matrix at energies above the two-particle threshold as an infinite series of separable terms, based on the Schmidt theorem, is presented. The Yamaguchi-type separable potential is assumed for the two-nucleon interaction. The application of this expansion to the calculation of a four-body resonance is studied by giving a numerical example. The convergence of the Schmidt expansion is comparable to that of the Hilbert-Schmidt expansion at energies below the two-particle threshold.

Separable pole expansions in four-nucleon bound state calculations

We compare the utility of the generalized unitary pole expansion and the energy-dependent pole expansion for the three-body subsystem amplitudes in four-body bound state calculations for a variety of separable and local nucleon-nucleon interactions. It is found that with the energy-dependent pole expansion the four-body binding energy is well reproduced with only two terms each for the (2 + 2) and the (3 + 1) subsystems, respectively, while the generalized unitary pole expansion requires three terms for the (3 + 1) channel and four terms for the (2 + 2) channel. We thus conclude that pole dominance is of greater importance for the generalized unitary pole expansion than for the energy-dependent pole expansion, which works equally well for both types of subsystems. It is found that both methods, in particular the energy-dependent pole expansion, converge more rapidly with increasing repulsion in the two-body interaction, i.e., the more realistic the interaction becomes. Both expansions require similar computing times for a converged calculation and are about 15-20 times faster than the widely used Hilbert-Schmidt expansion. The respective merits of the two pole expansions are discussed and compared with the Hilbert-Schmidt expansion.NUCLEAR STRUCTURE Four-body bound state calculations for various nucleon-nucleon interactions. Comparison of the generalized unitary pole expansion and energy-dependent pole expansion.

He4tetramer and the Faddeev-Yakubovsky theory

The Faddeev-Yakubovsky theory is applied to the system of ${(^{4}\mathrm{He})}_{4}$ interacting pairwise through a realistic He-He potential. The $S$-wave two-body $t$ matrix is obtained using the unitary pole expansion method, while the $S$-wave [3+1] and [2+2] subamplitudes are obtained by the Hilbert-Schmidt and the energy-dependent pole expansion (EDPE) methods. The convergences of the latter subamplitudes are tested and the superiority of the EDPE over the Hilbert-Schmidt expansion (HSE) is confirmed. The present results also confirm the results of a previous calculation using the ATMS method (amalgamation of two-body correlations into the multiple-scattering process).

Energy-dependent pole expansions for the effective potentials in the four-body integral equations with tensor forces

We investigate the accuracy of the energy-dependent pole expansion for the (3 + 1) and (2 + 2) subamplitudes in the calculation of the binding energy of the $\ensuremath{\alpha}$ particle, ${E}_{\ensuremath{\alpha}}$, for separable $\mathrm{NN}$ potentials with tensor components. We employ the truncated $t$-matrix (${t}_{00}$) approximation and compare our results for ${E}_{\ensuremath{\alpha}}$ to those obtained, independent of any separable expansion, by Gibson and Lehman and to the results for ${E}_{\ensuremath{\alpha}}$ obtained with the Hilbert-Schmidt expansion of the subamplitudes. It is shown that the energy-dependent pole expansion is both more economical and converges faster than the Hilbert-Schmidt expansion, even one term of the energy-dependent pole approximation already being accurate to better than 1.5%.NUCLEAR STRUCTURE Accuracy of energy-dependent pole expansion for calculation of $\ensuremath{\alpha}$-particle binding energy, compared to Hilbert-Schmidt expansion. Separable nucleon-nucleon potentials with tensor forces. Truncated $t$-matrix approximation.

Energy-dependent pole expansions for the effective potentials in the four-body integral equations

Energy-dependent pole expansions to the effective energy-dependent (3 + 1) and (2 + 2) transition amplitudes in the four-body integral equations are introduced. This separable expansion based on the pole dominance assumption is shown to be superior to the Hilbert-Schmidt expansion in its accuracy in the first approximation, in the number of terms required for convergence and in reducing the computational time by an order of magnitude.

Transition operators for atom–atom potentials: The Hilbert–Schmidt expansion

Transition operators 〈p‖t (ε) ‖p′〉 with p≠p′≠√2mε, required in collisions of atoms with bound atoms, are investigated and calculated within the Hilbert–Schmidt expansion. We have expanded in the eigenvalues and eigenfunctions of the kernel of the Lippmann–Schwinger equation, to obtain t operators for a Morse potential describing the ground 1Σ potential of H2, and for a repulsive Hulthén potential representing the lowest 3Σ potential of H2. Studies made at negative energies ε, required for inelastic and reactive processes below the dissociation threshold of the target, show that the expansion for the Morse potential is rapidly convergent, and that the relative magnitude of the + operator for the Hulthén potential is at least 1 order of magnitude smaller. Results are presented for momenta and energies appropriate to reactive collisions at energies just above the saddle-point energy of the H3 potential surface.

Application of the method of integral equations for calculating the vertex constants for an α-particle

The vertex constants G 2 αTN and G 2 αdd for the virtual decays α → t(τ) + p(n) and α → d + d are calculated by solving the Faddeev-Yakubovsky equations for four nucleons. A spin-dependent separable potential with the Hulthén form factor is used as the NN interaction. The resonant Hilbert-Schmidt expansion is applied to solve the integral equations. The values obtained G 2 αTN = 17.9 ± 1.7 fm and G 2 αdd = 18.1 ± 1.3 fm are compared with the phenomenological values extracted from data on nuclear reactions.

Low energy nucleon-trinucleon scattering in the integral equation approach

The n- 3He and n- 3H scattering processes are studied below the d-d breakup with the application of the Hilbert-Schmidt expansion in the Yakubovsky equations. As two-nucleon interactions are used s-wave local potentials of the Yukawa type.

Bound states of 4He with local interactions

The Yakubovsky four-body equations for the 0 +-state are solved in the boundstate region for the case of local central forces of the Yukawa type, using the Hilbert-Schmidt expansion. As a result the groundstate energy of 4He is found to be −29.6 MeV.

The integral equation approach to four-nucleon bound states

Within the framework of the Yakubovsky four-body equations the 0 + bound states of 4He are determined. The two-particle interactions used are of the separable Yamaguchi type and include spin-dependent forces. The problem is reduced to the solution of four coupled integral equations in two variables. The separable approximation of the kernels makes it possible to reduce the problem to a set of single variable integral equations. The separable approximation method employed is based on the Hilbert-Schmidt expansion applied to the kernels of four-body equations. The ground state energy of 4He is found to be −45.73 MeV, the excited 0 + level lies at −11.69 MeV. In conclusion we discuss the accuracy of various approximate methods in the four-nucleon problem.

The three-nucleon problem with the square-well potential

On the basis of the separable expansion for the two-particle t-matrix (the Hilbert-Schmidt expansion) the problem of three nucleons, the interaction between which is described by the central square-well potential, is considered. The triton binding energy and the neutron-deuteron scattering lengths are calculated.